Problem: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$7.50$ each for teachers and $$4.50$ each for students, and the group paid $$54.00$ in total. The next month, the same group visited a science museum where the tickets cost $$22.50$ each for teachers and $$10.00$ each for students, and the group paid $$137.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4.5y = 54}$ ${22.5x+10y = 137.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-22.5x-13.5y = -162}$ ${22.5x+10y = 137.5}$ Add the top and bottom equations together. $ -3.5y = -24.5 $ $ y = \dfrac{-24.5}{-3.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7.5x+4.5y = 54}$ to find $x$ ${7.5x + 4.5}{(7)}{= 54}$ $7.5x+31.5 = 54$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {22.5x+10y = 137.5}$ and get the same answer for $x$ ${22.5x + 10}{(7)}{= 137.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.